*Originally posted by phgao*

**Can you provide proof? And what about the 2nd Q?**

hmmm...actually, i think i made a mistake earlier. I think ABC should be 40 for the first problem.

My proof of the first question (I think this is right):

We are given that AC = XC. Therefore, it follows (law of sines, for example) that CAX = AXC. (Here my notation is that CAX refers to angle CAX)

So let CAX = AXC = x.

Also, let the radius of the circle be R.

It follows from the law of sines that XC/[sin(x)] = 2R = XC/[sin(XYC)].

It also follows from law of sines that AY/[sin(x+40)] = 2R = AY/[sin(ACY)].

We conclude from these two that:

sin(x) = sin(XYC) and

sin(x+40) = sin(ACY).

This does not mean that x = XYC and x+40 = ACY. In fact, these cannot both hold because then ABC would be 0 which doesn't make sense.

So the only solution I see that is in keeping with YXC = 40 is

XYC = 180-x and ACY = x+40. (for x < 90).

Then to find x, we just set the sum of angles within the circle to 360:

XAC + ACY + XYC + YXA = 360

-> x + (x+40) + (180-x) + (x+40) = 360

-> x = 50

Then it follows that ABC = 180 - XAC - ACY

-> ABC = 180 - (X) - (X+40) = 180 - 50 - 90 = 40

For the second problem, I was getting confused...I kept getting that the problem was not constrained, but I take it that it is, so I need to look at that problem again...